Properties

Label 2-177-1.1-c5-0-37
Degree $2$
Conductor $177$
Sign $-1$
Analytic cond. $28.3879$
Root an. cond. $5.32803$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 6.70·2-s + 9·3-s + 12.9·4-s + 105.·5-s − 60.3·6-s − 129.·7-s + 127.·8-s + 81·9-s − 704.·10-s − 188.·11-s + 116.·12-s − 958.·13-s + 866.·14-s + 945.·15-s − 1.27e3·16-s + 82.6·17-s − 543.·18-s − 2.06e3·19-s + 1.35e3·20-s − 1.16e3·21-s + 1.26e3·22-s − 589.·23-s + 1.14e3·24-s + 7.90e3·25-s + 6.42e3·26-s + 729·27-s − 1.67e3·28-s + ⋯
L(s)  = 1  − 1.18·2-s + 0.577·3-s + 0.404·4-s + 1.87·5-s − 0.684·6-s − 0.997·7-s + 0.705·8-s + 0.333·9-s − 2.22·10-s − 0.469·11-s + 0.233·12-s − 1.57·13-s + 1.18·14-s + 1.08·15-s − 1.24·16-s + 0.0694·17-s − 0.395·18-s − 1.31·19-s + 0.759·20-s − 0.575·21-s + 0.556·22-s − 0.232·23-s + 0.407·24-s + 2.52·25-s + 1.86·26-s + 0.192·27-s − 0.403·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $-1$
Analytic conductor: \(28.3879\)
Root analytic conductor: \(5.32803\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 177,\ (\ :5/2),\ -1)\)

Particular Values

\(L(3)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 9T \)
59 \( 1 - 3.48e3T \)
good2 \( 1 + 6.70T + 32T^{2} \)
5 \( 1 - 105.T + 3.12e3T^{2} \)
7 \( 1 + 129.T + 1.68e4T^{2} \)
11 \( 1 + 188.T + 1.61e5T^{2} \)
13 \( 1 + 958.T + 3.71e5T^{2} \)
17 \( 1 - 82.6T + 1.41e6T^{2} \)
19 \( 1 + 2.06e3T + 2.47e6T^{2} \)
23 \( 1 + 589.T + 6.43e6T^{2} \)
29 \( 1 + 1.12e3T + 2.05e7T^{2} \)
31 \( 1 - 940.T + 2.86e7T^{2} \)
37 \( 1 - 575.T + 6.93e7T^{2} \)
41 \( 1 + 5.52e3T + 1.15e8T^{2} \)
43 \( 1 + 1.65e4T + 1.47e8T^{2} \)
47 \( 1 - 2.38e4T + 2.29e8T^{2} \)
53 \( 1 + 3.96e4T + 4.18e8T^{2} \)
61 \( 1 - 1.26e3T + 8.44e8T^{2} \)
67 \( 1 + 4.91e4T + 1.35e9T^{2} \)
71 \( 1 + 5.36e4T + 1.80e9T^{2} \)
73 \( 1 - 3.08e4T + 2.07e9T^{2} \)
79 \( 1 + 2.67e4T + 3.07e9T^{2} \)
83 \( 1 - 2.74e4T + 3.93e9T^{2} \)
89 \( 1 + 3.01e4T + 5.58e9T^{2} \)
97 \( 1 + 8.02e4T + 8.58e9T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.55225162732665706250833349653, −9.951333908960479385483251026461, −9.471902601280208532707496788191, −8.591705720806519242341054520059, −7.26630241658312684052542800639, −6.25460421434292782479908963040, −4.84258066796979917162579184057, −2.68499442050990804705934193820, −1.79010961321683077222608869498, 0, 1.79010961321683077222608869498, 2.68499442050990804705934193820, 4.84258066796979917162579184057, 6.25460421434292782479908963040, 7.26630241658312684052542800639, 8.591705720806519242341054520059, 9.471902601280208532707496788191, 9.951333908960479385483251026461, 10.55225162732665706250833349653

Graph of the $Z$-function along the critical line